Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#9

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Question

A water tank holds 2000 litres when full. A small hole in the base is gradually getting bigger so that  each day a greater amount of water is lost.

     i.       On the first day after filling, 10 litres of water are lost and this increases by 2 litres each day. 

        a.   How many litres will be lost on the 30th day after filling?

        b.   The tank becomes empty during the nth day after filling. Find the value of n.

   ii.       Assume instead that 10 litres of water are lost on the first day and that the amount of water lost  increases by 10% on each succeeding day. Find what percentage of the original 2000 litres is left in the tank at the end of the 30th day after filling.

Solution


i.
 

From the given information, we can compile following data;

a.
 

The amount on first day;

The amount on 2nd day;

The amount on 3rd day;

We can sum up amounts as;

It is evident that it represents Arithmetic Progression (A.P);

The expression for difference  in Arithmetic Progression (A.P) is:

For amount of water Arithmetic Progression;

Since we are required to find the total amount of water lost on 30th day;

Expression for the general term  in the Arithmetic Progression (A.P) is:

b.
 

It is evident that tank will become empty on the day when sum of amounts of water lost day by day  becomes equal to total amount of water held by tank ie 2000 litres.

Expression for the sum of  number of terms in the Arithmetic Progression (A.P) is:

We recognize that it is a quadratic equation;

Standard form of quadratic equation is;

Solution of this quadratic equation is;

Therefore;

Now we have two options.

The days are more than 40.

NUMBER OF DAYS CANNOT
BE NEGATIVE

Hence, the tank will become empty on 41st day.


ii.
 

From the given information, we can compile following data for Model 1;

The amount on first day;

 

The amount on 2nd day;

The amount on 3rd day;

We can sum up water amounts as;

It is evident that it represents Geometric Progression (G.P);

Expression for Common Ratio () in a Geometric Progression (G.P) is;

To find the total amount of water lost till 30th day ;

Expression for the sum of  number of terms in the Geometric Progression (G.P) when  is:

Hence, 1645 litres water is lost till 30th day out of 2000 litres.

Total (2000-1645=)355 litres water is left in the tank.

Hence, 17.75% water is left in the day on 30th day.

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